14.2. STARTING THE L 2 (2) CASE OF .Cf EMPTY 987as J2 has two classes of involutions, cc;,(t) is isomorphic to a Sylow 2-subgroup of
the centralizer in Aut(J 2 ) of a non-2-central involution of J 2. Hence Cr(t) = A(k),
where A := CrnK(t) ~ E15 and k is an involution acting freely on A. Next
as a is parabolic isomorphic to the amalgam of Aut(J 2 ), there is j E T - K with
Z2 xD15 ~ Cr(j) E Syb(CKr(j)). Now Na(Cr(j)) normalizes f21((Cr(J))) = Z
and hence lies in Ca(z) =KT; it follows that Cr(j) is Sylow in Ca(j)-for other-
wise Cr(J) < X E Syb(Ca(j)) so that Cr(J) < Nx(Cr(j)) ::::; CKr(J), contrary
to Cr(J) E Syb(CKr(J)). But Cr(j) does not contain a copy of Cr(t) or T, so
ja nK = 0. Therefore by Thompson Transfer, j ¢:. 02 ( G), contrary to the simplicity
ofG.
Therefore a is not parabolic isomorphic to the Aut(J 2 )-amalgam, so a is par-
abolic isomorphic to the J2-amalgam, and K = Ca(z). G is of type J 2 or J 3 ,
completing the proof of (1). Then (2) and (3) follow from existing classification
theorems which we have stated in Volume I as I.4.7. D
In view of 14.2.17, to complete the proof of Theorem 14.2.7, it remains to treat
the^3 D 4 (2)-case. So assume a is the^3 D 4 (2) amalgam. Let Z = (z), G :=^3 D 4 (2),
and G := Aut(G).
LEMMA 14.2.18. Assume a is the^3 D 4 (2)-amalgam. Then Mc = Ca(z) and
either(1) Mc= K, or
(2) Mc= KA, where A::::; McnM is of order3 and induces field automorphisms
on K/Q. Moreover a:= (Mc, Mc n M, M) is the G-extension of a, in the sense of
Definition F.4.3.
PROOF. By 14.2.1.2, Mc = Ca(z). By 14.2.16, Q = 02(K) = 02(Mc), so
Mc/Q is faithful on Q by A.1.8. Now K E £(Mc, T) with K/02(K) ~ L2(8),
and T/Q ~ E 8 is Sylow in Mc/Q, so we conclude from 1.2.4 and A.3.12 that
K E C(Mc)· Then K :::1 Mc by 1.2.1.3 since T ::::; K. As the normalizer in
GL(Q) of K/Q is isomorphic to Aut(K/Q), either (1) holds or Mc/Q ~ Aut(L 2 (8)),
and we may assume the latter. Thus Mc = KA where A ::::; Na(T) is of order
3 and induces field automorphisms on K/Q. Then A acts on CQ(T) = V, so
A::::; Na(V) = M. As Mc= KA, Mn Mc= B1A, so M = Y(M n Mc) =YT A.
Then a:= (Mc, Mn Mc, M) satisfies Hypothesis F.1.1 just as a did, and hence by
F.1.9, a is a weak EN-pair of rank 2. Then a is an extens.ion of its sub-amalgam
a, which we have already identified; so a is the G-extension of a. D
LEMMA 14.2.19. If a is the^3 D 4 (2) amalgam, then G ~^3 D 4 (2).
PROOF. Let ry :=a in case (1) of 14.2.18, and ry :=a in case (2) of 14.2.18. In
either case, by 14.2.18, ry is an extension of the^3 D4(2)-amalgam, with the role of ·
"G 1 " played by Mc= Ca(z). Thus the hypotheses of Theorem F.4.31 are satisfied
since G = 02 (G), so by that Theorem, G is an extension of^3 D 4 (2) of odd degree,
and hence isomorphic to^3 D4(2) since G is simple. D
Observe that 14.2.17 and 14.2.19 establish Theorem 14.2.7.
14.2.2. The treatment of certain cases where H is solvable. We next
analyze the case where for some HE 'H*(T, M), H/0 2 (H) is either a group of Lie
rank 1 over F2 isomorphic to L2(2) or Sz(2) ~ F20, or H/0 2 (H) ~ D10. We do